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Class 10th Maths || Menu
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  • Surface Areas and Volumes
    • Ex. 13.1 NCERT
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CBSE Class 10 Maths Chapter 13 Surface Areas and Volumes Notes

Surface Areas and Volumes Class 10 Notes Maths Chapter 13 1

SURFACE AREA AND VOLUME OF COMBINATIONS

Cone on a Cylinder.
Surface Areas and Volumes Class 10 Notes Maths Chapter 13 2
r = radius of cone & cylinder;
h1 = height of cone
h2 = height of cylinder
Total Surface area = Curved surface area of cone + Curved surface area of cylinder + area of circular base
= πrl + 2πrh2 +πr2;
Slant height, l = \sqrt { { r }^{ 2 }+{ { { h }_{ 1 }^{ 2 } } } }
Total Volume = Volume of cone + Volume of cylinder
= \frac { 1 }{ 3 } { \pi r }^{ 2 }{ h }_{ 1 }+{ \pi r }^{ 2 }{ h }_{ 2 }

Cone on a Hemisphere:
Surface Areas and Volumes Class 10 Notes Maths Chapter 13 3
h = height of cone;
l = slant height of cone = \sqrt { { r }^{ 2 }+{ h }^{ 2 } }
r = radius of cone and hemisphere
Total Surface area = Curved surface area of cone + Curved surface area of hemisphere = πrl + 2πr2
Volume = Volume of cone + Volume of hemisphere = \frac { 1 }{ 3 } { \pi r }^{ 2 }h+\frac { 2 }{ 3 } { \pi r }^{ 3 }

Conical Cavity in a Cylinder
Surface Areas and Volumes Class 10 Notes Maths Chapter 13 4
r = radius of cone and cylinder;
h = height of cylinder and conical cavity;
l = Slant height
Total Surface area = Curved surface area of cylinder + Area of bottom face of cylinder + Curved surface area of cone = 2πrh + πr2 + πrl
Volume = Volume of cylinder – Volume of cone = { \pi r }^{ 2 }h-\frac { 1 }{ 3 } { \pi r }^{ 2 }h=\frac { 2 }{ 3 } { \pi r }^{ 2 }h

Cones on Either Side of Cylinder.
Surface Areas and Volumes Class 10 Notes Maths Chapter 13 5
r = radius of cylinder and cone;
h1 = height of cylinder
h2 = height of cones
Slant height of cone, l = \sqrt { { h }_{ 2 }^{ 2 }+{ r }^{ 2 } }
Surface area = Curved surface area of 2 cones + Curved surface area of cylinder = 2πrl + 2πrh1
Volume = 2(Volume of cone) + Volume of cylinder = \frac { 2 }{ 3 } { \pi r }^{ 2 }{ h }_{ 2 }+{ \pi r }^{ 2 }{ h }_{ 1 }

Cylinder with Hemispherical Ends.
Surface Areas and Volumes Class 10 Notes Maths Chapter 13 6
r = radius of cylinder and hemispherical ends;
h = height of cylinder
Total surface area= Curved surface area of cylinder + Curved surface area of 2 hemispheres = 2πrh + 4πr2
Volume = Volume of cylinder + Volume of 2 hemispheres = { \pi r }^{ 2 }h+\frac { 4 }{ 3 } { \pi r }^{ 3 }

Hemisphere on Cube or Hemispherical Cavity on Cube
Surface Areas and Volumes Class 10 Notes Maths Chapter 13 7
a = side of cube;
r = radius of hemisphere.
Surface area = Surface area of cube – Area of hemisphere face + Curved surface area of hemisphere
= 6a2 – πr2 + 2πr2 = 6a2 + πr2
Volume = Volume of cube + Volume of hemisphere = { a }^{ 3 }+\frac { 4 }{ 3 } { \pi r }^{ 3 }

Hemispherical Cavity in a Cylinder
Surface Areas and Volumes Class 10 Notes Maths Chapter 13 8
r = radius of hemisphere;
h = height of cylinder
Total surface area = Curved surface area of cylinder + Surface area of base + Curved surface area of hemisphere
= 2πrh + πr2 + 2πr2 = 2πrh + 3πr2
Volume = Volume of cylinder – Volume of hemisphere = { \pi r }^{ 2 }h-\frac { 2 }{ 3 } { \pi r }^{ 3 }

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